Regularization of Toda lattices by Hamiltonian reduction

Laszlo Feher; Izumi Tsutsui

doi:10.1016/S0393-0440(96)00010-1

paper

Regularization of Toda lattices by Hamiltonian reduction

arXiv:hep-th/9511118 · doi:10.1016/S0393-0440(96)00010-1

Abstract

The Toda lattice defined by the Hamiltonian $H = \frac{1}{2} \sum_{i = 1}^{n} p_{i}^{2} + \sum_{i = 1}^{n - 1} ν_{i} e^{q_{i} - q_{i + 1}}$ with $ν_{i} \in {\pm 1}$ , which exhibits singular (blowing up) solutions if some of the $ν_{i} = - 1$ , can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is $T^{*} G_{e}$ , where $G_{e} = N_{+} A N_{-}$ consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group $N_{+} \times N_{-}$ . Using the Bruhat decomposition we show that the full reduced system obtained from $T^{*} G$ , which is perfectly regular, contains $2^{n - 1}$ Toda lattices. More precisely, if $n$ is odd the reduced system contains all the possible Toda lattices having different signs for the $ν_{i}$ . If $n$ is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ${\Real}^{2n-1}$. If $ν_{i} = 1$ for all $i$ , we prove for $n = 2, 3, 4$ that the Toda phase space associated with $T^{*} G_{e}$ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.

42 pages, plain TeX, one reference added, to appear in J. Geom. Phys

Regularization of Toda lattices by Hamiltonian reduction

Abstract

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